That was some good troubleshooting you did to find the problem. If your regulator is built into the alternator, you will have to replace the alternator.

One thing you can check is to make sure none of the smaller wires going to the alternator have power. I think the one you are looking for in particular is a red/green wire if I remember right. This comes from the keyswitch, and when it has voltage, it brings the alternator "online". Of course if this wire had power all the time, the alternator would be online all the time and run the battery down.

If the wiring checks out good, it's probably something wrong in the circuit in the regulator that triggers the alternator field windings to activate.